Analysis

Real analysis

Basic concepts

• Laplace transform, Fourier transform, Mellin transform
• functional dependence
• entire function
• Riemann zeta function
• Padé aproximant
• integration and derivation
• Lipschitz continuous function and uniformly continuous function
• Weierstrass theorem.
• Bessel functions.
• Jacobi elliptic function.
• Bolzano-Weierstrass theorem.
• Hessian matrix
• integration on Rn

Numerical methods

• Runge-Kutta

Differential equations

• famous equations
• Green's function method
• reduction of order for linear ODEs
• lowering order by chain rule

Functional analysis

• Banach space
• Hilbert space
• spectral theory
• symmetric operator
• Hermitian matrix and unitary matrix.
• variational problem
• extrema
• distribution (functional analysis)
• Stone's theorem
• convolution (discrete)

Complex analysis

• holomorphic function
• Cauchy--Goursat theorem
• Cauchy integral formula
• Riemann zeta function
• meromorphic function
• Hartog's theorem

%%{ init: { "themeVariables": { "fontSize": "10px" } } }%%
graph TD
    A[Functions of a Complex Variable] --> B[Holomorphic Functions]
    B --> C[Cauchy–Riemann Conditions]
    B --> D[Differentiability and Analyticity]
    D --> E[Power Series]
    E --> M[Laurent Series]
    M --> N[Singularities]
    N --> O[Residue theorem]
    B --> F[Integration in the Complex Plane]
    F --> G[Cauchy's Theorem]
    G --> H[Cauchy's Integral Formula]
    H --> I[Consequences: Derivatives; Cauchy's Inequality]
    I --> J[Morera's Theorem]
    J --> K[Liouville's Theorem & Maximum Modulus Principle]
    K --> L[Open Mapping Theorem]
    A --> Q[Conformal Transformations]
    Q --> R[Möbius Transformations]